Polarization-dependent circular Dammann grating
made of azo-dye-doped liquid crystals
Dan Luo,
1Xiao Wei Sun,
1,2,* Hai Tao Dai,
2and Hilmi Volkan Demir
1,3,4 1School of Electrical and Electronic Engineering, Nanyang Technological University,Nanyang Avenue, Singapore 639798, Singapore
2
Department of Applied Physics, College of Science, Tianjin University, Tianjin 300072, China
3School of Physical and Mathematical Sciences, Nanyang Technological University,
Nanyang Avenue, Singapore 639798, Singapore
4Department of Electrical and Electronics Engineering and Department of Physics,
Bilkent University, Bilkent, Ankara, 06800, Turkey *Corresponding author: exwsun@ntu.edu.sg
Received 28 October 2010; revised 2 March 2011; accepted 30 March 2011; posted 31 March 2011 (Doc. ID 137203); published 19 May 2011
A polarization-dependent circular Dammann grating (CDG) was generated from an azo-dye-doped liquid crystal (LC) cell. A simple multiexposure photo-alignment process was used to fabricate a binary phase LC CDG zone plane, which was composed of an odd zone with a twisted nematic LC structure and an even zone with a homogenous LC structure. A two-order CDG with equal-intensity rings was produced through a Fourier transform. The maximum zeroth and first diffraction orders of obtained CDG can be separately achieved by rotating the analyzer’s polarization direction. The CDG using an azo-dye-doped LC cell can be used to generate diffractions by lasers in a broad wavelength range, hence expanding possible device applications. © 2011 Optical Society of America
OCIS codes: 230.3720, 050.1970, 110.5220.
1. Introduction
A circular Dammann grating (CDG) is a diffraction grating that generates a circular equal-intensity pat-tern at the image plane [1]. Circular Dammann grat-ings with periodic structures have been designed using the Hankel transform method [2,3] and the cir-cular spot rotation method [4]. The phase modulation of the CDG can be binary [2–4], which is the simplest type, multilevel [5], or continuous. As a useful dif-fractive optical element, the CDG was reported to be a useful measurement tool for quick determina-tion of the back focal length of a lens [6]. The separa-tion of rings in the focal length of a CDG can be used in collimation testing of a laser beam [7]. Wen et al. reported the practical application of a CDG in
angular and area measurement, where angles and areas can be rapidly obtained with maintained accu-racy [8,9]. In addition, the CDG was also used in an optical image coding system [10].
Azo dye is a kind of photosensitive material that is capable of aligning liquid crystal (LC) molecules through a photoisomerization molecular reorienta-tion process [11,12]. Azo-dye-doped nematic LC is a good candidate for fabricating polarization-related optical devices, such as polarization Fresnel lenses [13,14], polarization gratings [15–17], vertical photo-nic crystals [18], polarization rotators [19], and photonic crystal fiber devices [20].
Previously reported CDGs were usually fabri-cated by optical-inactive materials (resist, glass etc.), suitable only for a single wavelength due to the re-quirement of wavelength-dependent π phase differ-ence for high diffractive efficiency. In consequdiffer-ence, the application of such CDGs is limited. In this paper,
0003-6935/11/152316-06$15.00/0 © 2011 Optical Society of America
a broad band polarization-dependent CDG was generated from an azo-dye-doped nematic LC cell. A binary phase LC two-order CDG zone plane con-sisting of an odd zone with a twisted nematic (TN) LC structure and an even zone with a homogenous LC structure was fabricated through a multiexpo-sure photoalignment process. The maximum zeroth and first/second diffraction orders of a CDG can be separately achieved by changing the analyzer’s direc-tion, thus leading to a polarization-dependent tun-able CDG. The CDG based on azo-dye-doped LC can be operated in a broad wavelength range due to the waveguide property of the TN LC structure.
A binary phase CDG, as a kind of radially periodic phase grating, can generate impulse rings with equal intensity in the far field. The number of the diffractive order with equal intensity in the spectral domain is defined as the order of a CDG. A radially periodic im-age gðrÞ with period T can be decomposed into a linear combination of circular sinusoid functions as [2]
gðrÞ ¼X∞ n¼1 cnsin 2πn Tr ; ð1Þ
where the nth term represents the nth ring. Its Four-ier spectrum calculated from FourFour-ier transform (or Hankel transform) is given by [2]
GðqÞ ¼ 1pffiffiffiπX∞
n¼1
cn n=T
ðn=T þ qÞ3=2δð1=2Þðq − n=TÞ; ð2Þ
where q¼ ρ=λf is the spatial frequency, with ρ repre-senting a polar coordinate in the observed plane and f representing the focal length of the lens, andδð1=2ÞðxÞ is the half-order derivative of the impulseδðxÞ. For the case of a two-order CDG with a profile as shown in Fig. 1(a), where fr1¼ 0:1634; r2¼ 0:5; r3¼ 0:8366; r4¼ 1g are the normalized phase transitions in a per-iod length and are chosen from [2] directly, the corre-sponding theoretical spectral intensity in the far field is presented in Fig. 1(b), and the theory diffraction efficiency limit is 65.2%. The calculated coefficients
cnare given by
cn¼ 2
nπ½−2 cosð2πnx1Þ þ ð−1Þ
nþ2þ 1; ð3Þ
where x1¼ r1¼ 0:1634. The designed two-order bin-ary phase CDG pattern was then transferred to a photomask with a period of T¼ 200 μm and a total aperture of 8 mm, as depicted in Fig.1(c).
2. Experiment and Simulation
The materials used herein consisted of 99 wt: % LC, E7 (no¼ 1:5216 and ne¼ 1:7462) from Merck, and
1 wt: % azo dye, Methyl Red (MR) from Aldrich. Two indium tin oxide (ITO) coated glass slides were used to assemble an empty cell with a gap of d¼ 30 μm. The dye-doped LC mixture was then infil-trated into an empty cell. As both glass substrates were not rubbed, the alignment of LC in the cell was initially random.
The LC molecules in the cell can be reoriented by linearly polarized light in the absorption band of azo dye. When excited by light in the absorption spec-trum, the azo dye in the LC sample would undergo trans–cis isomerization, followed by molecular reor-ientation, diffusion, and adsorption on the ITO glass surface. The adsorbed dyes then align liquid crystals by forcing the LC molecules to reorient along the direction of the azo dye’s long axis [13,14].
In our experiment, a diode-pumped linearly polar-ized solid state laser light at a wavelength of 532 nm, which is in the absorption band of MR dye (410–550 nm, with its peak around 500 nm), was used to irradiate the dye-doped LC sample for photo-aligning LC molecules in the cell. The absorption of MR dye at 532 nm is high and is about 70% of that at peak absorption wavelength of 500 nm. The photoa-lignment process for fabricating a CDG on the azo-dye-doped LC sample consists of two steps. In the first step, as shown in Figs. 2(a)–2(c), glass sub-strates 1 and 2 were illuminated by the laser light with an intensity level of 120 mW=cm2for 3 min with
linear polarization along the x axis separately. Then the MR dye on the substrate aligned the LC to be per-pendicular to the incident light polarization, leading to a parallel alignment along the y axis of the LC on both substrates and, therefore, to the formation of a homogeneous LC layer structure. In the second step, glass substrate 2, covered by the photomask with a CDG pattern, was illuminated by laser with linear polarization along the y axis. The odd circle/rings of the CDG are transparent, and the even rings are opaque, as shown in Fig.2(d). During the second illumination, the MR dye on substrate 2 reoriented the LC again, and a TN structure for the odd zone was finally achieved, while the homogenous struc-ture of the even zone remained unchanged, as shown in Fig. 2(e). The adsorption-induced twist angle, defined as the angle between the original LC direc-tor’s direction and the reoriented LC molecule direc-tor’s by azo dye, is dependent on the exposure time
Fig. 1. (Color online) (a) Cross section of a two-order binary phase ð0; πÞ CDG, (b) theoretical normalized intensity of a two-order CDG, (c) two-order binary phase CDG pattern transferred onto a photomask.
and pumping intensity of illumination light [19]. In our experiment, for demonstrating a more general case, a 60° TN structure for the odd circle/rings of the CDG zone was obtained using a laser intensity of 120 mW=cm2and exposure time of 2 min for the
sec-ond illumination in Fig.2(d). A larger twist angle TN structure (e.g., 90° TN) could be obtained by length-ening the exposure time or increasing the illumina-tion intensity. Such a multiexposure photoalignment process provides a simple way to align the LC in the cell, as no rubbing is needed.
For the odd zone of the cell with θ° TN structure (whereθ is the twist angle between the LC directors on substrate 1 and 2) as presented in Fig.2(e), if light with linear polarization along the y axis (parallel to the LC director on substrate 1) is incident from the substrate 1 side, theθ° TN structure will rotate the polarization of light by ∼θ° after it passes through the cell when Mauguin’s condition is satis-fied (Δnd ≫ λ, where Δn ¼ ne− no and λ is
wave-length of incident light) [21], which means the polarization of output light passing through the odd zone will be rotated to be parallel to the LC di-rector on substrate 2. In our experiment, Mauguin’s condition was satisfied and θ was equal to 60°. For the even zone with homogeneous structure, the polarization of incident light would not be changed after it passes through the cell, when light with polarization along the y axis is incident from substrate 1.
When the LC cell sample was examined under an optical microscope with two polarizers, the white light source would pass through polarizer 1 (P1), sub-strate 1, LC cell, subsub-strate 2, and polarizer 2 (P2) in sequence in our experiment. The direction of polari-zer 1 was set to be parallel to the LC director on substrate 1, both along the y axis. The obtained mi-croscope images of the photoaligned azo-dye-doped LC with a CDG pattern are shown in Fig.3, where Figs. 3(a) and 3(b) represent images captured with
polarizer–polarizer angles of 30° and 90°, corre-sponding to a completely dark odd zone and a completely dark even zone respectively. The direc-tions of polarizer 1 and polarizer 2 for each case are drawn below each image. For the odd zone, a completely darkened image appeared only when polarizer 2 had a 30° angle with polarizer 1 as shown in Fig. 3(a), indicating that the LC director (D) on substrate 2 should be perpendicular to polarizer 2 and has a 60° angle with the y axis, as shown in
Fig. 2. (Color online) Schematic fabrication process of azo-dye-doped LC CDG.
Fig. 3. (Color online) Optical microscope images of the photoa-ligned azo-dye-doped LC cell with CDG pattern in the cases of (a) 30° angle of polarizer–polarizer and (b) crossed polarizer– polarizer. Solid line and dashed line represent polarizer 1 and po-larizer 2, respectively. The scale bars are 200 μm. (c) The directions of LC directors on substrates 1 and 2 are shown for odd and even ring zones, respectively.
the left part of Fig.3(c). Therefore, a 60° TN structure can be confirmed in the odd circle/ring zone. Simi-larly, for the even ring zone, a completely darkened image appeared only when polarizer 2 was perpendi-cular to polarizer 1, as shown in Fig.3(b), confirming a parallel alignment LC structure.
For generating CDG diffraction, the azo-dye-doped LC cell sample recorded with a CDG pattern was il-luminated by a collimated He-Ne laser (633 nm) with a beam diameter of about 1 cm and intensity of 0:2 mW=cm2 to generate the CDG. The absorption
of MR at 633 nm is very low and is around 10% of that at peak absorption wavelength of 500 nm. A polarizer was used to ensure that the polarization of incident light was parallel to the LC director on substrate 1 (both along the y axis). A spherical lens (f ¼ 20 cm) placed right behind the LC cell sample was used to perform the Fourier transform of the CDG pattern. An analyzer was used to verify the image polariza-tion. The circular diffraction pattern was obtained at the focal plane of the Fourier-transform lens and then captured by a CCD camera.
3. Results and Discussions
Figure4shows the polarization-dependent property of the azo-dye-doped LC CDG recorded by CCD. The polarization of the incident light, parallel to the LC
directors on substrate 1 in the even zone, would not change, because the even zone has a homogeneous LC structure, while the polarization of the incident light through the odd zone is rotated by∼60°, because the odd zone has 60° TN LC structure. That is called the waveguiding effect [21] in the TN LC structure. As a result, the angle between the polarizations of output light from the even and odd zones is around 60°. Figure4presents the observed images recorded by a CCD with (a) no analyzer, (b) 30°, (c) 15°, (d) 0°, (e)−15°, (f) −30°, (g) −45°, and (h) −60° of analyzer– polarizer angle, where 0° is along the y axis. Corre-sponding output light polarization from the even (Eeven) and odd (Eodd) zones, polarization (P), and analyzer (A) directions are indicated below each image, respectively.
Figure 4(a) shows the observed image obtained without using an analyzer. We can see that both the zeroth and first/second diffraction orders corre-sponding to the light spot and CDG two equal-intensity rings, respectively, existed in this case. Figures4(b)and4(h)show the observed images with the maximum zeroth and first/second diffraction orders, where the analyzer was set to 30° and −60°, respectively, compared to the polarizer direction (y axis). The reason behind this observed image can be found as follows. Assuming the incident light
Fig. 4. (Color online) Observed images recorded by a CCD (a) without an analyzer and with (b) 30°, (c) 15°, (d) 0°, (e) −15°, (f) −30°, (g) −45°, and (h) −60° analyzer–polarizer angle, where 0° is along the y axis. Corresponding output light polarization from even (Eeven) and odd (Eodd) zones, polarization (P), and analyzer (A) directions are shown below each image.
has a polarization state parallel to 0°, after passing through the CDG, the polarization of output light in the even zone remained unchanged, while that in the odd zone became parallel to 60°, defined in Fig. 4. When the analyzer was set along the direction of 30°, defined in Fig. 4, the electric field components of⇀Eevenand E
⇀
odd along the direction of the analyzer
are equal to each other, with both the same ampli-tude and phase, leading to a zero phase difference of output light from the even and odd zones and fi-nally the zeroth diffraction order maximum. In con-trast, when the analyzer was set along−60°, defined in Fig.4, the electric field components of⇀Eeven and
E
⇀
odd along the direction of analyzer have the same
amplitude but a π phase difference, leading to a π phase difference of output light from the even and odd zones and finally the desired CDG image with the first/second diffraction order maximum. If the analyzer was set between 30° and −60°, the obtained images would be in one of the intermediate states with both zeroth and first/second orders, and these corresponding images are given in Figs. 4(c)–4(g). Therefore, by adjusting the analyzer’s direction, we can gradually change the output CDG image, which makes it possible to use it as a polarization-dependent tunable optical element.
From Fig.4(h), we can see that two equal-intensity rings are produced from our sample as predicted; the unwanted zero order in the CDG image was due to its high sensitivity to fabrication errors, where a very bright zero order can be produced even when a small phase error exists [2]. However, this did not prevent us from demonstrating the utility of the polarization-dependent azo-dye-doped LC CDG. The splitting of the main intensity maximum is due to the property of the binary phase CDG and could be eliminated by using a four-phase-level CDG [2].
Figure 5 shows the normalized intensity of the two-order binary CDG given in Fig.4(h), whereρ re-presents the distance from the center of CDG in the polar coordinates. The splitting of the main intensity maximum, as discussed previously by S. Zhao et al. [2,6], is due to the property of binary phase CDG and could be eliminated by a four-phase-level CDG [2]. The radii of produced intensity rings can be calcu-lated from q¼ ρ=λf ¼ n=T, and the theoretical radius of the first and second order rings on the im-age plane areρ1¼ λf =T ¼ 633 μm and ρ2¼ 2λf =T ¼ 1266 μm, respectively, where λ ¼ 633 nm, f ¼ 20 cm, and T¼ 200 μm in our experiment. The obtained experimental ρ1 and ρ2 were around 606 μm and 1263 μm, respectively, slightly smaller than what is predicted, which was due to the errors of the radii’s positions in the binary phase CDG [2]. Therefore, our experimental results are quite consistent with the simulation.
It is worth noting that any laser 1) satisfying Mauguin’s condition and 2) that is sufficiently far away from MR azo-dye absorption peak, or even near the absorption peak but then with low intensity, could be used to generate the polarization-dependent
CDG diffraction due to the light-rotating (waveguid-ing) property of TN structure in the odd zone. This broadband property expands the CDG application range, which is of great advantage compared to previously reported CDGs operating at only one fixed working wavelength.
4. Conclusion
In conclusion, we have generated a polarization-dependent CDG diffraction from an azo-dye-doped LC cell. A simple multiexposure photoaligned pro-cess was used to fabricate a binary phase LC CDG zone plane that consisted of an even zone with a homogenous LC structure and an odd zone with a 60° TN LC structure. The maximum zeroth and first/ second diffraction orders of the CDG can be sepa-rately achieved by changing the analyzer’s polariza-tion direcpolariza-tion. If the chosen analyzer’s direction leads to a zero phase difference in output light from the even and odd zones, the maximum zeroth diffraction order will be achieved. In contrast, if the chosen ana-lyzer’s direction leads to a π phase difference in out-put light from the even and odd zones, the maximum first diffraction order will be produced. The CDG dif-fraction from our azo-dye-doped LC cell can be gen-erated by lasers within a broad wavelength band, which expands possible applications.
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